Long-time behavior of the solution to the mKdV equation with step-like initial data

We consider the modified Korteweg-de Vries equation q(t) + 6q(2)q(x) + q(xxx) = 0 on the line. The initial data is the pure step function, i.e. q(x, 0) = c(r) for x >= 0 and q(x, 0) = c(l) for x < 0, where c(l) > c(r) > 0 are arbitrary real numbers. Long-time behavior of the solution to the mKdV equation in the case c(l) > c(r) = 0 and t -> infinity was studied recently in Kotlyarov and Minakov (2010 J. Math. Phys. 51 093506) where the structure of the compression wave was obtained in the form of a modulated elliptic wave. The goal of this paper is to study the asymptotic behavior of the solution of the initial-value problem as t -> -infinity, i.e. we study the long-time dynamics of the rarefaction wave. Using the steepest descent method and the so-called g-function mechanism we deform the original oscillatory matrix Riemann-Hilbert problem to the explicitly solvable model forms and show that the solution of the initial-value problem has a different asymptotic behavior in different regions of the xt-plane. In the regions x < 6c(l)(2)t and x > 6c(r)(2)t the main term of asymptotics of the solution is equal to c(l) and c(r), respectively. In the region 6c(l)(2)t < x < 6c(r)(2)t the asymptotics of the solution tends to root x/6t. An influence of the dispersion is also studied: the second term of the asymptotics is obtained in the region x > -6c(r)(2)t, where the background constant c(r) is perturbed by the self-similar vanishing wave.